This will probably depend on one's political affiliation. I think that a case can be made that the Great Society really strove to embody the very best of America. The notion that a nation cannot be great unless it is meeting the needs of all of its citizens, from the very best to the most destitute, is a powerfully compelling idea. The desire to want to provide education to those who are in need and display want, the idea of ensuring that the most feeble of a social order is guaranteed health care, or the idea that America can eliminate poverty are all extremely praiseworthy initiatives. In this light, one could say that measures conceived with such ideas in mind would be able to improve the quality of life for many in American Society.
The Great Society was far more than just the War on Poverty. Many programs did aid the impoverished, and while the upper and middle classes may not see a direct benefit from that, the enlarged tax base by raising people out of poverty impacts everyone in the long run.
Beyond poverty, the Great Society addressed the injustices of racisim by pushing through such legislastion as the Civil Rights Act of 1964 and the Voting Rights Act of 1965. Money was given to universities to create scholarships and low-interest loans. Medicare and Medicaid were designed to provide medical assistance to the elderly and the poor. There are other programs that were funded as well in the arts.
Most of what Lyndon Johnson did through the Great Society was meant to help poor people. So a lot of people who are not poor got no direct benefits from the Great Society.
For example, some of the most high profile programs were Head Start, which is preschool for poor kids, Medicaid, which is health care for poor people, and general welfare programs, which are, of course, for poor people.
So LBJ helped improve the lives (you can argue) of poor people through programs like the three I just mentioned. I would say that most middle class people were not directly affected.